Question: In one online store, $8\%$ of the orders are placed by new customers. Let $N$ be the number of orders placed in a day to reach the store's first new customer. Assume each order's customer status is independent. Find the probability that the first new customer places the $5^{\text{th}}$ order of the day. You may round your answer to the nearest hundredth. $P(N=5)=$
Answer: Without a fancy calculator For each order: $P({\text{new}})=0.08$ $P(\text{old}})=0.92$ If the $5^{\text{th}}$ order is the first one placed by a new customer, the sequence of customers must be "old, old, old, old, new." $\begin{aligned} P(N=5)&=P(\text{old}}, \text{old}}, \text{old}}, \text{old}}, {\text{new}}) \\\\ &=(0.92})(0.92})(0.92})(0.92})({0.08}) \\\\ &=(0.92)^4(0.08) \\\\ &\approx0.0573 \end{aligned}$ $P(N=5) \approx 0.0573 \approx 0.06$